direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D4.S3, D20.7D6, Dic6⋊9D10, C60.7C23, Dic30⋊2C22, C3⋊C8⋊15D10, C3⋊6(D5×SD16), D4.7(S3×D5), (D4×D5).1S3, (C5×D4).1D6, D4.D15⋊1C2, (D5×Dic6)⋊1C2, (C3×D5)⋊2SD16, (C6×D5).61D4, (C4×D5).44D6, C6.140(D4×D5), C15⋊10(C2×SD16), C15⋊3C8⋊5C22, (C3×D4).18D10, C30.169(C2×D4), C30.D4⋊2C2, C6.D20⋊2C2, C20.7(C22×S3), C12.7(C22×D5), (C3×Dic5).12D4, (C5×Dic6)⋊2C22, (C3×D20).3C22, (D4×C15).1C22, (D5×C12).3C22, D10.39(C3⋊D4), Dic5.12(C3⋊D4), (D5×C3⋊C8)⋊2C2, C4.7(C2×S3×D5), C5⋊2(C2×D4.S3), (C3×D4×D5).1C2, (C5×C3⋊C8)⋊5C22, (C5×D4.S3)⋊1C2, C2.22(D5×C3⋊D4), C10.43(C2×C3⋊D4), SmallGroup(480,559)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D4.S3
G = < a,b,c,d,e,f | a5=b2=c4=d2=e3=1, f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >
Subgroups: 716 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C2×SD16, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×C3⋊C8, D4.S3, D4.S3, C2×Dic6, C6×D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×D4.S3, C5×C3⋊C8, C15⋊3C8, D5×Dic3, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, Dic30, D4×C15, D5×C2×C6, D5×SD16, D5×C3⋊C8, C30.D4, C6.D20, C5×D4.S3, D4.D15, D5×Dic6, C3×D4×D5, D5×D4.S3
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C3⋊D4, C22×S3, C2×SD16, C22×D5, D4.S3, C2×C3⋊D4, S3×D5, D4×D5, C2×D4.S3, C2×S3×D5, D5×SD16, D5×C3⋊D4, D5×D4.S3
►On 120 points
Generators in S120
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 49)(2 50)(3 46)(4 47)(5 48)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(1 79 19 64)(2 80 20 65)(3 76 16 61)(4 77 17 62)(5 78 18 63)(6 81 21 66)(7 82 22 67)(8 83 23 68)(9 84 24 69)(10 85 25 70)(11 86 26 71)(12 87 27 72)(13 88 28 73)(14 89 29 74)(15 90 30 75)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)
G:=sub<Sym(120)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120), (1,49)(2,50)(3,46)(4,47)(5,48)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,66,71)(62,67,72)(63,68,73)(64,69,74)(65,70,75)(76,81,86)(77,82,87)(78,83,88)(79,84,89)(80,85,90)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120), (1,79,19,64)(2,80,20,65)(3,76,16,61)(4,77,17,62)(5,78,18,63)(6,81,21,66)(7,82,22,67)(8,83,23,68)(9,84,24,69)(10,85,25,70)(11,86,26,71)(12,87,27,72)(13,88,28,73)(14,89,29,74)(15,90,30,75)(31,106,46,91)(32,107,47,92)(33,108,48,93)(34,109,49,94)(35,110,50,95)(36,111,51,96)(37,112,52,97)(38,113,53,98)(39,114,54,99)(40,115,55,100)(41,116,56,101)(42,117,57,102)(43,118,58,103)(44,119,59,104)(45,120,60,105)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79)(82,85)(83,84)(87,90)(88,89)(92,95)(93,94)(97,100)(98,99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119), (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120), (1,49)(2,50)(3,46)(4,47)(5,48)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,66,71)(62,67,72)(63,68,73)(64,69,74)(65,70,75)(76,81,86)(77,82,87)(78,83,88)(79,84,89)(80,85,90)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120), (1,79,19,64)(2,80,20,65)(3,76,16,61)(4,77,17,62)(5,78,18,63)(6,81,21,66)(7,82,22,67)(8,83,23,68)(9,84,24,69)(10,85,25,70)(11,86,26,71)(12,87,27,72)(13,88,28,73)(14,89,29,74)(15,90,30,75)(31,106,46,91)(32,107,47,92)(33,108,48,93)(34,109,49,94)(35,110,50,95)(36,111,51,96)(37,112,52,97)(38,113,53,98)(39,114,54,99)(40,115,55,100)(41,116,56,101)(42,117,57,102)(43,118,58,103)(44,119,59,104)(45,120,60,105) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(62,65),(63,64),(67,70),(68,69),(72,75),(73,74),(77,80),(78,79),(82,85),(83,84),(87,90),(88,89),(92,95),(93,94),(97,100),(98,99),(102,105),(103,104),(107,110),(108,109),(112,115),(113,114),(117,120),(118,119)], [(1,49,19,34),(2,50,20,35),(3,46,16,31),(4,47,17,32),(5,48,18,33),(6,51,21,36),(7,52,22,37),(8,53,23,38),(9,54,24,39),(10,55,25,40),(11,56,26,41),(12,57,27,42),(13,58,28,43),(14,59,29,44),(15,60,30,45),(61,91,76,106),(62,92,77,107),(63,93,78,108),(64,94,79,109),(65,95,80,110),(66,96,81,111),(67,97,82,112),(68,98,83,113),(69,99,84,114),(70,100,85,115),(71,101,86,116),(72,102,87,117),(73,103,88,118),(74,104,89,119),(75,105,90,120)], [(1,49),(2,50),(3,46),(4,47),(5,48),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(61,66,71),(62,67,72),(63,68,73),(64,69,74),(65,70,75),(76,81,86),(77,82,87),(78,83,88),(79,84,89),(80,85,90),(91,96,101),(92,97,102),(93,98,103),(94,99,104),(95,100,105),(106,111,116),(107,112,117),(108,113,118),(109,114,119),(110,115,120)], [(1,79,19,64),(2,80,20,65),(3,76,16,61),(4,77,17,62),(5,78,18,63),(6,81,21,66),(7,82,22,67),(8,83,23,68),(9,84,24,69),(10,85,25,70),(11,86,26,71),(12,87,27,72),(13,88,28,73),(14,89,29,74),(15,90,30,75),(31,106,46,91),(32,107,47,92),(33,108,48,93),(34,109,49,94),(35,110,50,95),(36,111,51,96),(37,112,52,97),(38,113,53,98),(39,114,54,99),(40,115,55,100),(41,116,56,101),(42,117,57,102),(43,118,58,103),(44,119,59,104),(45,120,60,105)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 5 | 5 | 20 | 2 | 2 | 10 | 12 | 60 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 6 | 6 | 30 | 30 | 2 | 2 | 8 | 8 | 4 | 20 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | SD16 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | D4.S3 | S3×D5 | D4×D5 | C2×S3×D5 | D5×SD16 | D5×C3⋊D4 | D5×D4.S3 |
| kernel | D5×D4.S3 | D5×C3⋊C8 | C30.D4 | C6.D20 | C5×D4.S3 | D4.D15 | D5×Dic6 | C3×D4×D5 | D4×D5 | C3×Dic5 | C6×D5 | D4.S3 | C4×D5 | D20 | C5×D4 | C3×D5 | C3⋊C8 | Dic6 | C3×D4 | Dic5 | D10 | D5 | D4 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D5×D4.S3 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 188 | 52 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 188 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 133 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 228 | 225 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 19 | 222 | 0 | 0 | 0 | 0 |
| 222 | 222 | 0 | 0 | 0 | 0 |
| 0 | 0 | 216 | 237 | 0 | 0 |
| 0 | 0 | 156 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,188,0,0,0,0,1,52],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,188,0,0,0,0,0,1],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,240,133,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,228,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[19,222,0,0,0,0,222,222,0,0,0,0,0,0,216,156,0,0,0,0,237,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D5×D4.S3 in GAP, Magma, Sage, TeX
D_5\times D_4.S_3
% in TeX
G:=Group("D5xD4.S3"); // GroupNames label
G:=SmallGroup(480,559);
// by ID
G=gap.SmallGroup(480,559);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^4=d^2=e^3=1,f^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations